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A generalized Exner equation for sediment mass balance

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 110, F04014, doi:10.1029/2004JF000274, 2005 A **generalized** **Exner** **equation** **for** **sediment** **mass** **balance** C. Paola 1 and V. R. Voller 2 St. Anthony Falls Laboratory, Minneapolis, Minnesota, USA Received 13 December 2004; revised 7 June 2005; accepted 6 September 2005; published 30 November 2005. [1] The advance of morphodynamics research into new areas has led to a proliferation of **for**ms of **sediment** **mass** **balance** **equation**. Without a general **equation** it is often difficult to know what these problem-specific versions of **sediment** **mass** **balance** leave out. To address this, we derive a general **for**m of the standard **Exner** **equation** **for** **sediment** **mass** **balance** that includes effects of tectonic uplift and subsidence, soil **for**mation and creep, compaction, and chemical precipitation and dissolution. The complete **equation**, (17), allows **for** independent evolution of two critical interfaces: that between bedrock and **sediment** or soil and that between **sediment** and flow. By eliminating terms from the general **equation** it is straight**for**ward to derive **mass** **balance** **equation**s applicable to a wide range of problems such as short-term bed evolution, basin evolution, bedrock uplift and soil **for**mation, and carbonate precipitation and transport. Dropping terms makes explicit what is not being considered in a given problem and can be done by inspection or by a **for**mal scaling analysis of the terms. Scaling analysis leads directly to dimensionless numbers that measure the relative importance of terms in the **equation**, **for** example, the relative influence of spatial versus temporal changes in **sediment** load on bed evolution. Combining scaling analysis with time averaging shows how the relative importance of terms in the **equation** can change with timescale; **for** example, the term representing bed evolution due to temporal change in **sediment** load tends to zero as timescale increases. Citation: Paola, C., and V. R. Voller (2005), A **generalized** **Exner** **equation** **for** **sediment** **mass** **balance**, J. Geophys. Res., 110, F04014, doi:10.1029/2004JF000274. 1. Introduction [2] An analysis of **sediment** **mass** **balance** is fundamental to solving a wide range of problems in morphodynamics. The relations in common use **for** expressing **sediment** **mass** **balance** take a variety of **for**ms but are all descended from the **equation** initially presented by **Exner** and reproduced below. The additions and changes that have been made to **Exner**’s original **equation** have mostly been done piecemeal with the aim of adapting it **for** a particular problem. A summary of the **Exner** **equation** including **for**ms appropriate **for** channelized systems and **sediment** mixtures is presented by Parker [2005], and detailed derivations of **mass** **balance** **for** soil-mantled hillslopes are presented by Anderson [2002] and Mudd and Furbish [2004]. Both of the latter works also explicitly include chemical effects. Nonetheless, each of these **mass** **balance** **equation**s includes some terms and leaves out others, according to the problem at hand. Here we revisit the **Exner** **equation** with the aim of providing a complete general **for**m. This generality allows the **equation** to be specialized **for** application to a wide range of morphodynamic problems by dropping or combining terms. 1 Also at Department of Geology and Geophysics, University of Minnesota Twin Cities, Minneapolis, Minnesota, USA. 2 Also at Department of Civil Engineering, University of Minnesota Twin Cities, Minneapolis, Minnesota, USA. Copyright 2005 by the American Geophysical Union. 0148-0227/05/2004JF000274$09.00 We hope that the proposed general **for**m will be especially useful **for** geologic problems **for** which processes such as tectonic uplift and subsidence, soil **for**mation and creep, and dissolution and precipitation become important. [3] It is easy to overlook the fact that a model is defined as much by what it leaves out as by what it includes. Using a general **equation** as a starting point makes this explicit: we develop specialized relations **for** specific problems by eliminating terms from the general **equation**. Elimination of terms can be done by inspection or by **for**mal scaling analysis. We give examples of how to estimate the magnitudes of different terms below, after deriving the general **mass** **balance** **equation**. Because surface evolution occurs on timescales from seconds to millions of years, we then investigate the behavior of the general **equation** under time averaging. We conclude the paper by returning to one of the original motivations **for** it: the satisfaction of finding unity in apparently diverse phenomena. We hope that the examples that conclude the paper, showing how the general **mass** **balance** **equation** can be specialized to a variety of problems from classical morphodynamics to basin dynamics and carbonate precipitation, will illustrate how **sediment** **mass** **balance** can serve as one such unifying theme in surface dynamics. 2. **Exner**’s Equation [4] Felix **Exner** was a Viennese meteorologist who worked on a variety of topics in natural science. He F04014 1of8

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